Telling people what we can and cannot do is an essential language skill. As my previous sentence shows we use modal verbs in English to express that ability. However, Korean doesn’t have ‘modal verbs’ as they exist in English. Instead they use grammatical verb endings to achieve the same effect. The verb ending to express ability is -(으)ㄹ 수 있다/없다.
Your Ability and Inability
You can use the verb ending (으)ㄹ 수 있다 to tell someone of what you can to do. You can also use this same expression to inform others when it is possible to do something or when something is possible. It is a fairly easy verb ending to understand as it has little hassle to it.
However, if you want to tell someone you cannot do something, something is impossible for you to do, or simply that something is impossible, you will need to use the verb ending -(으)ㄹ 수 없다. I believe that the use of 있다/없다 makes the difference in meaning between the two quite clear.
Furthermore it goes without saying, but to make sure it is clear, this verb ending is generally only used with action verbs, not adjectives (=descriptive verbs).
수 + 가
It is possible to add even more emphasis on the ability or possibility you are talking about, if you feel it is needed. To do so you will need the subject particle 가. You simply attach 가 to the word 수 of this verb ending to add added emphasis. In this case 가 is akin to saying ‘really’ (정말).
The How To
Using the forms -(으)ㄹ 수 있다/없다 with a verb isn’t that hard either. First you will need the verb stem, which is quite easy. Then you simply have to slap the verb ending to it. When the verb ending ends in a vowel you will need to use ㄹ 수 있다/없다. When the verb stem ends in a consonant (batchim), you will slap -을 수 있다/없다 at the end. Of course, be aware of the Korean irregular, some irregular verbs tend to be quirky with this verb ending.